In General Relativity we see spacetime as a manifold; in this context vectors can't be defined on the manifold but need to be defined on the tangent space of the manifold. So each point of the manifold has its own tangent space and different vectors in different tangent spaces cannot be easily compared. At last each tangent space has its own metric tensor $g_{\mu \nu}=\partial _\mu \cdot \partial _\nu$, where $\partial _\mu,\partial _\nu$ are the base of the tangent space.
Problem is: my geometrical intuition makes me think about the tangent space as a flat space; if you have any 2D or 3D object in everyday experience the tangent space at one point is always a flat one. But it's not only intuition: spacetime locally looks like $\mathbb{M}^4$, so locally it looks flat or to say it better: locally can be approximated with a flat spacetime; but seems to me that the tangent space at one point is simply the space that better approximate the area around that point; this also push me to say that the tangent space should be always flat.
So is the tangent space of a manifold always flat? Or equivalently is the tangent space always $\mathbb{M}^4$?
Based on the upper reasoning seems to me that the answer should be yes, but this seems to create a problem: in GR we use the Christoffel connection so the curvature can be calculated using only the metric tensor $g_{\mu\nu}$, but if the tangent space is always flat then the metric tensor is always "a flat one", in the sense that it generates always flat curvature. This is obviously absurd. How can we get out of this apparent contradiction?
Edit: Based on the answer of Javier tangent space is indeed always flat. Does it mean that I can take any tangent space (with metric tensor $g_{\mu\nu}$), apply a change of coordinates and get the metric tensor of $\mathbb{M}^4$ ($\eta _{\mu\nu}$)? This is important because this is what flatness means; am I right?
And also: we state that the metric is calculated by looking at the rate of change of the metric $g_{\mu\nu}=\partial _\mu \cdot \partial _\nu$, but the metric is always flat! So the rate of change is always zero because every tangent space is flat! How can we deal with this?
Best Answer
Short answer: yes each tangent space is flat.
In principle the tangent space is a vector space, not a Riemannian manifold, and so the concept of curvature technically doesn't apply if that's all you have. To define curvature you need to define parallel transport; for that, you need to think of the tangent space as a manifold, and that implies looking at the tangent spaces of the tangent space! And if you do that, the vector space structure gives you a canonical way to define parallel transport, and this parallel transport ends up being flat.
Finally, the solution to your paradox is simple: the curvature depends on derivatives of the metric, that is, on how it changes from tangent space to tangent space. The metric at a point is irrelevant because all vector spaces with an inner product are isometric; what matters is how it varies in space.
Edit in response to your edit: You're playing too fast and loose with the words. Differential geometry is complicated, and we need to be precise in how we speak.
You can always make any tangent space into Minkowski space by choosing an appropriate basis, not because it is flat, but because it is a vector space. The difference is subtle but important. A vector space has a single metric tensor: it takes pairs of vectors and returns a number. A manifold has a metric tensor field: a metric tensor at each point, which takes pairs of tangent vectors. The fact that the tangent space is flat is a red herring.
One definition of flatness (of a manifold) is that you can use a single coordinate system in which the metric tensor is everywhere Minkowski. This statement is different from the statement about the tangent space; here you're choosing a different basis at each point, which are related by coming from a single coordinate system. In a single tangent space, you only have one basis. And you can always have $g_{\mu\nu} = \eta_{\mu\nu}$ at a single point (and hence at a single tangent space), but making it so at every point with the same coordinate system may or may not be possible, and that's what flatness means.